Available at:http://www.pmf.ni.ac.rs/filomat
Non Self-Adjoint Laplacians on a Directed Graph
Marwa Baltia
aUniversit´e de Carthage, Facult´e des Sciences de Bizerte: Math´ematiques et Applications (UR/13ES47) 7021-Bizerte (Tunisie) Universit´e de Nantes, Laboratoire de Math´ematique Jean Lauray, CNRS, Facult´e des Sciences, BP 92208, 44322 Nantes, (France).
Abstract.We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights.
We analyse spectral properties of this Laplacian under a Kirchhoffassumption. Moreover we establish isoperimetric inequalities in terms of the numerical range to show the absence of the essential spectrum of the Laplacian onheavy enddirected graphs.
Introduction
The non self-adjoint operators are more difficult to study than the self-adjoint ones: no spectral theorem in general, wild resolvent growth... The related theory is studied by different authors: L. N. Trefethen [18]
for non symmetric matrices, W. D. Evans, R. T. Lewis, A. Zettl [7] and R. T. Lewis [16] for non self-adjoint operators in a Hilbert space. Recently, the interest in spectral properties of non self-adjoint operators has already led to a variety of new results, both in the continuous and discrete settings, e.g, bounds on complex eigenvalues [8] and Lieb-Thirring type inequalities [12], [4]. This can be explained by the complicated structure of the resolvent of such an operator seen as an analytic function. In this paper we focus on directed graphs to study a non symmetric Laplacian. We develop a general approximation theory for the eigenvalues on directed graphs with non symmetric edge weights assuming only a condition of ”total conductivity of the vertices” presented as the Assumption (β). We investigate the spectrum of our discrete non self-adjoint Laplacian. We collect some basic properties of the Laplacian and we seek to show the emptiness of its essential spectrum by using isoperimetric inequalities. We explain how isoperimetric inequalities can be linked to the numerical range of non symmetric operators. In fact, for the self-adjoint Laplace-Beltrami operator, JeffCheeger proved an inequality that links the first nontrivial eigenvalue on a compact Riemannian manifold to a geometric constanth. This inspired an analogous theory for graphs (see [9], [10]). In this work, we introduce a kind of Cheeger constant on a filtration of a directed graphG and we estimate the associated Laplacian∆. We give an estimation for the numerical range of∆in terms of the Cheeger constant. We use this estimation and propose a condition on the weights for the absence
2010Mathematics Subject Classification. 47A45, 47A12, 47A10, 47B25
Keywords. Directed graph, Graph Laplacian, Non self-adjoint operator, Cheeger constant, Numerical range, Eigenvalues, Essential spectrum.
Received: 02 December 2016; Accepted: 09 July 2017 Communicated by Paola Bonacini
This work was financially supported by the ”PHC Utique” program of the French Ministry of Foreign Affairs and Ministry of higher education and research and the Tunisian Ministry of higher education and scientific research in the CMCU project number 13G1501 ”Graphes, G´eom´etrie et th´eorie Spectrale”. Also I like to thank the Laboratory of Mathematics Jean Leray of Nantes (LMJL) and the research unity (UR/13ES47) of Faculty of Sciences of Bizerta (University of Carthage) for their financial and their continuous support.
Email address:balti-marwa@hotmail.fr(Marwa Balti)
of essential spectrum ofheavy enddirected graphs. There is an analogous result of H. Donnelly and P. Li [6] for a self-adjoint operator on complete negatively curved manifolds. They show that the Laplacian on a rapidly curving manifold has a compact resolvent.
Section 1 is devoted to some definitions and notions on a directed graph with non symmetric edge weights and the associate non symmetric differential Laplacian. We describe some basic results: Green’s formula and the spectral properties of∆and of its formal adjoint.
In Section 2, we study spectral properties of the bounded operator ˜∆by relying on known results for the symmetric case.
In Section 3, we establish the Cheeger inequality for the non symmetric Dirichlet Laplacian on any subset of the set of verticesVto give a lower bound for the bottom of the real part of the numerical range. We control the real part of the numerical range of∆and relate it with the spectrum of the its closure∆. We characterize the absence of essential spectrum of∆. Fujiwara [9] and Keller [15] introduced a criterion for the absence of essential spectrum of the symmetric Laplacian on a rapidly branching graph. In fact, our criterion is: positivity of the Cheeger constant at infinity onheavy endgraphs.
1. Preliminaries
We review in this section some basic definitions on infinite weighted graphs and introduce the notation used in the article. They are introduced in [2] for finite non symmetric graphs (see [1] and [17] for the symmetric case).
1.1. Notion of Graphs
A directed weighted graph is a tripletG:=(V, ~E,b), whereVis a countable set (the vertices),E~is the set of directed edges andb:V×V→[0,∞) is a weight function satisfying the following conditions:
• b(x,x)=0 for allx∈V
• b(x,y)>0 iff(x,y)∈E~
In addition, we consider a measure on V given by a positive function m:V→(0,∞).
The weighted graph issymmetricif for allx,y∈V,b(x,y)=b(y,x), as a consequence (x,y)∈~E⇔(y,x)∈E.~ The graph is calledsimpleif the weightsmandbare constant and equal to 1 onVandE~respectively.
The setEof undirected edges is given by E=n
{x,y},(x,y)∈E~or (y,x)∈E~o .
Definition 1.1. Define for a subsetΩof V, the vertex boundary and the edge boundary ofΩrespectively by:
∂VΩ =n
y∈Ω: {x,y} ∈E for some x∈Ωco
∂EΩ =n
(x,y)∈E~: (x∈Ω, y∈Ωc) or (x∈Ωc, y∈Ω)o . On a non symmetric graph we have two notions of connectedness.
Definition 1.2. • A path between two vertices x and y in V is a finite set of directed edges(x1,y1); (x2,y2);..; (xn,yn),n≥ 2such that
x1=x, yn=y and xi=yi−1 ∀2≤i≤n
• G is called connected if two vertices are always related by a path.
• G is called strongly connected if for all vertices x,y there is a path from x to y and one from y to x.
1.2. Functional spaces
Let us introduce the following spaces associated to the graphG:
• the space of functions on the graphGis considered as the space of complex functions on V and is denoted by
C(V)={f :V→C}
• Cc(V) is its subset of finite supported functions;
• we consider for a measurem, the space
`2(V,m)={f ∈ C(V), X
x∈V
m(x)|f(x)|2<∞}.
It is a Hilbert space when equipped by the scalar product given by (f,1)m=X
x∈V
m(x)f(x)1(x).
The associated norm is given by:
kfkm= p (f,f)m. 1.3. Laplacian on a directed graph
In this work, we assume that the graph under consideration is connected, locally finite, without loops and satisfies for allx∈Vthe following conditions:
X
y∈V
b(x,y)>0 and X
y∈V
b(y,x)>0.
We introduce the combinatorial Laplacian∆defined onCc(V) by:
∆f(x)= 1 m(x)
X
y∈V
b(x,y) f(x)− f(y).
For allx∈Vwe note byβ+(x)=X
y∈V
b(x,y), in particular ifm(x)=β+(x) then the Laplacian is said to be the normalized Laplacian and it is defined by:
∆˜f(x)= 1 β+(x)
X
y∈V
b(x,y)
f(x)−f(y) .
Dirichlet operator: LetUbe a subset ofV, f ∈ Cc(U) and1:V→Cthe extension of f toVby setting 1=0 outsideU. For any operatorAonCc(V), the Dirichlet operatorADUis defined by
ADU(f)=A(1)|U.
The operator∆may be non symmetric if the edge weight is not symmetric.
Proposition 1.3. The formal adjoint∆0of the operator∆is defined onCc(V)by:
∆0f(x)= 1 m(x)
X
y∈V
b(x,y)f(x)−X
y∈V
b(y,x)f(y)
.
Proof. For all f,1∈ Cc(V), we have (∆f,1)m= X
(x,y)∈~E
b(x,y)
f(x)− f(y) 1(x)
=X
x∈V
f(x)X
y∈V
b(x,y)1(x)− X
(y,x)∈~E
b(y,x)1(y)f(x)
=X
x∈V
f(x)
X
y∈V
b(x,y)1(x)− X
(y,x)∈~E
b(y,x)1(y)
.
As (∆f,1)m=(f,∆01)m, so we get
∆0f(x)= 1 m(x)
X
y∈V
b(x,y)f(x)−X
y∈V
b(y,x)f(y)
.
Remark 1.4. The operator∆0can be expressed as a Schr¨odinger operator with the potential q(x)= 1
m(x) X
y∈V
b(x,y)−b(y,x)
, x∈V:
∆0f(x)= 1 m(x)
X
y∈V
b(y,x)
f(x)−f(y)
+q(x)f(x).
We introduce here theAssumption(β) and we assume that it is satisfied by the considered weighted graph, throughout the rest.
Assumption(β): for allx∈V, β+(x)=β−(x) where
β+(x)=X
y∈V
b(x,y) andβ−(x)=X
y∈V
b(y,x).
Remark 1.5. The Assumption(β)is natural. Indeed, it looks like the Kirchhoff’s law in the electrical networks.
Corollary 1.6. We suppose that the Assumption(β)is satisfied, the operator∆0is simply a Laplacian, given by
∆0f(x)= 1 m(x)
X
y∈V
b(y,x)
f(x)− f(y) .
In the sequel, for the sake of simplicity we introduce the symmetric LaplacianHassociated to the graph with the symmetric edge weight functiona(x,y)=b(x,y)+b(y,x). It acts onCc(V) by,
H f(x)=(∆ + ∆0)f(x)= 1 m(x)
X
y∈V
a(x,y)
f(x)−f(y) .
The quadratic formQ∆ofHis given by
Q∆(f)=(∆f,f)+(∆f,f), f ∈ Cc(V).
Comment 1.7. Let f ∈ Cc(V), we have Q∆(f)=2Re(∆f,f). Then
kfinfkm=1Q∆(f)= inf
kfkm=12Re(∆f,f). (1)
We establish an explicit Green’s formula associated to the non self-adjoint Laplacian∆from which any estimates on the symmetric quadratic formQ∆can be directly cited from the literature.
Lemma 1.8. (Green’s Formula) Let f and1be two functions ofCc(V). Then under the Assumption(β)we have (∆f,1)m+(∆1,f)m= X
(x,y)∈~E
b(x,y)
f(x)− f(y)
1(x)−1(y) .
Proof. The proof is given by a simple calculation. From Corollary 1.6, we have (∆f,1)m+(∆1,f)m=(H f,1)m
= X
(x,y)∈~E
b(x,y)
f(x)− f(y)
1(x)+ X
(y,x)∈~E
b(y,x)
f(x)− f(y) 1(x)
= X
(x,y)∈~E
b(x,y)
f(x)1(x)+ f(x)1(x)− f(y)1(x)− f(x)1(y)
= X
(x,y)∈~E
b(x,y)
f(x)− f(y)
1(x)−1(y) .
We refer to [14] page 243 for the definitions of the spectrum and the essential spectrum of a closed operatorAin a Hilbert spaceH, with domainD(A).
Definition 1.9. • The spectrumσ(A)of A is the set of all complex numbersλsuch that(A−λ)has no bounded inverse.
• The essential spectrumσess(A)of A is the set of all complex numbersλfor which the range R(A−λ)is not closed ordim ker(A−λ)=∞.
2. Spectral Analysis of the Bounded Case
This part concerns some basic properties of the bounded non self-adjoint Laplacian ˜∆. We introduce the concept of the numerical range. It has been extensively studied the last few decades. This is because it is very useful in studying and understanding the spectra of operators (see [3], [13], [2]).
Definition 2.1. The numerical range of an operator T with domain D(T), denoted by W(T)is the non-empty set W(T)={(T f,f), f ∈D(T), k f k=1}.
The following Theorem in [13] shows that the spectrum behave nicely with respect to the closure of the numerical range.
Theorem 2.2. LetHbe a reflexive Banach space and T a bounded operator onH. Then:
σ(T)⊂W(T).
The following Proposition is one of the main tools when working with the normalized Laplacian.
Proposition 2.3. Suppose that the Assumption(β)is satisfied. Then∆˜ is bounded by 2.
Proof.
|( ˜∆f,1)β+|=|X
x∈V
1(x)X
y∈V
b(x,y)
f(x)−f(y)
|
≤X
x∈V
β+(x)|f(x)1(x)|+X
x∈V
|1(x)|X
y∈V
b(x,y)| f(y)|
(2) by the Assumption (β) and the Cauchy-Schwarz inequality we prove the result:
|( ˜∆f,1)β+|≤
f,f12
β+
1,112
β++X
x∈V
|1(x)| X
y∈V
b(x,y)12 X
y∈V
b(x,y)| f(y)|212
≤ f,f12
β+(1,1)
1
β2++ X
x∈V
β+(x)|1(x)|212 X
x∈V
X
y∈V
b(x,y)| f(y)|212
≤(f,f)
1
β2+(1,1)
1
β2++(1,1)
1
β2+
X
y∈V
| f(y)|2X
x∈V
b(x,y)12
≤ f,f12
β+
1,112
β++ 1,112
β+
X
y∈V
| f(y)|2β−(y)12
≤2 f,f12
β+
1,112
β+. Then
k∆˜kβ+= sup
kfkβ+≤1 k1kβ+≤1
|( ˜∆f,1)β+ |≤2.
It is useful to develop some basic properties of the numerical range to make the computations of the spectrum of the Laplacian.
Proposition 2.4. Let G be a connected graph, satisfying the Assumption(β). Then 1. σ( ˜∆)⊂D(1,1), the closed disc with center(1,0)and radius1.
2. Ifβ+(V)<∞, then0is a simple eigenvalue of∆.˜
Proof. 1. By Cauchy-Schwarz inequality as in (2), forf ∈D(∆) we have
|( ˜∆f,f)β+−(f,f)β+ |=|X
x∈V
X
y∈V
b(x,y)f(x)f(y)|
≤X
x∈V
X
y∈V
b(x,y)| f(x)f(y)|
≤(f,f)β+ which implies thatW( ˜∆)⊂D(1,1).
2. IfX
x∈V
β+(x) = X
x∈V
β+(x) < ∞, the constant function is an eigenfunction of ˜∆ associated to 0. Then 0 is an eigenvalue of ˜∆. Now, we suppose that f is an eigenfunction of ˜∆ associated to 0, therefore ( ˜∆ +∆˜0)f,f
=0. Thus by connectedness ofG, f is constant.
It is obvious thatRe(A)= 1 2
A+A∗
ifAis a bounded operator, but this is not true in general. The result below establishes a link between the real part of a matrix and its eigenvalues considered as the roots of the characteristic polynomial, see [11] page 8. The adjoint of a square matrix is the transpose of its conjugate.
Lemma 2.5. Let A be a square matrix of size n,λk(A)andλk(Re(A)), k =1, ..,n the eigenvalues of A andRe(A) respectively. Suppose that the eigenvalues of Re(A) are labelled in the increasing order, so that, λ1(Re(A)) ≤ λ2(Re(A))..≤λn(Re(A)). Then
Xn
k=n−q+1
Re(λk(A))≤ Xn
k=n−q+1
λk(Re(A)), ∀q=1, ..,n
and the equality prevails for q=n.
Remark 2.6. It should be noted that for a matrix A,λk(Re(A))andRe(λk(A))are not equal in general. We can see [2] for a counter-example.
In the following we study some generalities of eigenvalues of ˜∆DΩ, whereΩ is a finite subset ofV. We assume that they are ordered as follows:
Re(λ1( ˜∆DΩ))≤ Re(λ2( ˜∆DΩ))..≤ Re(λn( ˜∆DΩ)).
Lemma 2.7. LetΩbe a finite non-empty subset of V, we have
λ1(Re( ˜∆DΩ))≤ Re(λ1( ˜∆DΩ)).
Proof. Let f be an eigenfunction associated toλ1( ˜∆DΩ). By the variational principle ofλ1( ˜HDΩ), we have λ1( ˜HDΩ)≤ ( ˜HDΩf,f)m
(f,f)m
= ( ˜∆DΩf,f)m
(f,f)m +( ˜∆DΩf,f)m (f,f)m
=λ1( ˜∆DΩ)+λ1( ˜∆DΩ).
The next statement contains an additional information about the eigenvalues of ˜∆DΩ.
Proposition 2.8. LetΩbe a finite non-empty subset of V(#Ω =n)such that∂VΩ,∅. Then the following assertions are true
1. 0<Re(λ1( ˜∆DΩ))≤1.
2. λ1(Re( ˜∆DΩ))+λn(Re( ˜∆DΩ))≤2.
Proof. 1. From Theorem 4.3 of [10], we haveλ1(Re( ˜∆DΩ)) > 0 and by Lemma 2.7 we conclude the left inequalty. Next, by Lemma 2.5 we have forq=n:
n
X
k=1
Re(λk( ˜∆DΩ))=
n
X
k=1
λk(Re( ˜∆DΩ) then
nRe(λ1( ˜∆DΩ))≤
n
X
k=1
λk(Re( ˜∆DΩ)=Tr
Re( ˜∆DΩ)
=n which proves that
Re(λ1( ˜∆DΩ))≤1.
2. It is deduced from the result of the symmetric case, see Theorem 4.3 [10].
Corollary 2.9. LetΩbe a finite non-empty subset of V, then Re(λn( ˜∆DΩ))<2.
Proof. Applying the Lemma 2.5 forq=1, we get
Re(λn( ˜∆DΩ))≤λn(Re( ˜∆DΩ)).
But by (2) of Proposition 2.8, we have :
λn(Re( ˜∆DΩ))≤2−λ1(Re( ˜∆DΩ)).
Then from the general propertyλ1(Re( ˜∆DΩ))>0, we conclude thatλn(Re( ˜∆DΩ))<2.
3. Spectral Study of the Unbounded Case
This part includes the study of the bounds on the numerical range and the essential spectrum of a closed Laplacian. Both issues can be approached via isoperimetric inequalities.
3.1. Closable operator
The purpose of the theory of unbounded operators is essentially to construct closed extensions of a given operator and to study their properties.
Definition 3.1. Closable operators: A linear operator T:D(T)→ His closable if it has closed extensions.
An interesting property for the Laplacian∆is its closability.
Proposition 3.2. Let G be a graph satisfying the Assumption(β). Then∆is a closable operator.
Proof. We shall use the Theorem of T. Kato which says that an operator densely defined is closable if its numerical range is not the whole complex plane, see [14], page 268. Letλ∈W(∆), there is f ∈ Cc(V) such thatk f km=1 andλ=(∆f,f)m. From the Green’s formula we have,
2Re(λ)= X
(x,y)∈~E
b(x,y)| f(x)− f(y)|2≥0.
It follows thatW(∆)⊂n
λ∈C, Re(λ)≥0o ( C.
For such operators, another property of interest is the property of being closed.
Definition 3.3. The closure of∆is the operator∆, defined by
• D(∆)=n
f ∈`2(V,m), ∃(fn)n∈N∈ Cc(V), fn→ f and ∆fnconvergeo
• ∆f := lim
n→∞∆fn, f ∈D(∆)and(fn)n∈ Cc(V)such that fn→ f .
For an unbounded operator the relation between the spectrum and the numerical range is more complicated.
But for a closed operator we have the following inclusion, see [14] and [2].
Proposition 3.4. Let T be a closed operator. Thenσess(T)⊂W(T).
More precisely, let us define the following numbers:
η(T)=inf{Reλ: λ∈σ(T)}. ν(T)=inf{Reλ: λ∈W(T)}. ηess(T)=inf{Reλ: λ∈σess(T)}. The Proposition 3.4 induces this Corollary.
Corollary 3.5.
ηess(∆)≥ν(∆). (3)
Remark 3.6. If∆is self-adjoint, thenη(∆)=ν(∆). But this is not the case in general.
3.2. Cheeger inequalities
For a non symmetric graphG, we prove bound estimates on the real part of the numerical range of∆in terms of the Cheeger constant. We use this estimation to characterize the absence of the essential spectrum of∆.
First, we recall the definitions of the Cheeger constants onΩ⊂V:
h(Ω) = inf
U⊂Ω f inite
b(∂EU) m(U) and
h(Ω)˜ = inf
U⊂Ω f inite
b(∂EU) β+(U) where for a subsetUofV,
b(∂EU)= X
(x,y)∈∂EU
b(x,y)
β+(U)=X
x∈U
β+(x) andm(U)=X
x∈U
m(x).
We define in addition:
mΩ=inf (β+(x)
m(x), x∈Ω )
MΩ=sup (β+(x)
m(x), x∈Ω )
.
Cheeger’s Theorems had appeared in many works on symmetric graphs. They give estimations of the bottom of the spectrum of the Laplacian in terms of the Cheeger constant. The inequality (4) controls the lower bound of the real part ofλ∈W(∆DΩ).
Theorem 3.7. LetΩ⊂V, the bottom of the real part of W(∆DΩ)satisfies the following inequalities:
h2(Ω)
8 ≤ MΩν(∆DΩ) ≤ 1
2MΩh(Ω). (4)
Proof. From the works of J. Dodziuk [5] and A. Grigoryan [10], we can deduce the following bounds of the symmetric quadratic formQ∆D
Ω onCc(Ω), h2(Ω)
8 ≤ MΩ inf
kfkm=1Q∆D
Ω(f) ≤ 1
2MΩh(Ω).
Then using the equality (1) we conclude our estimation.
We deduce in particular the following inequalities.
Corollary 3.8. LetΩ⊂V, we have h˜2(Ω)
8 ≤ ν( ˜∆DΩ) ≤ 1 2h(Ω).˜
Proposition 3.9. LetΩ⊂V and1∈ Cc(Ω), k1km=1. Letλ=(∆DΩ1,1)m∈W(∆DΩ). Then mΩ
Re( ˜∆DΩ1,1)β+
(1,1)β+ ≤2Re(λ)≤MΩ
Re( ˜∆DΩ1,1)β+
(1,1)β+ . (5)
Proof. We have for allx∈Ω
mΩm(x)≤β+(x)≤MΩm(x) therefore
mΩ(1,1)m≤(1,1)β+ ≤MΩ(1,1)m
which implies that:
mΩ
Re( ˜∆DΩ1,1)β+ (1,1)β+ ≤ Q∆D
Ω(1) 2(1,1)m
≤MΩ
Re( ˜∆DΩ1,1)β+ (1,1)β+ because (∆DΩ1,1)m=( ˜∆DΩ1,1)β+, for all1∈ Cc(Ω).
Corollary 3.10. LetΩ⊂V, we have mΩh˜2(Ω)
8 ≤ ν(∆DΩ). (6)
We can also estimate the real part of any element of the numerical range of∆DΩin terms of the isoperimetric constant ˜h.
Corollary 3.11. For allΩ⊂V andλ∈W(∆DΩ)we have mΩ
2− q
4−h˜2(Ω)
≤2Re(λ)≤MΩ 2+ q
4−h˜2(Ω)
. (7)
Proof. We follow the same approach as Fujiwara in Proposition 1 [9], and we apply it to the symmetric Laplacian ˜HDΩ=∆˜DΩ+∆˜0DΩ, we obtain, for all1∈ Cc(Ω)
2− q
4−h˜2(Ω)≤ 2Re( ˜∆DΩ1,1)β+ (1,1)β+
≤2+ q
4−h˜2(Ω).
Hence we obtain the result by a direct corollary of the inequality (5).
3.3. Absence of essential spectrum from Cheeger constant
This subsection is devoted to the study of the essential spectrum relative to the geometry of the weighted graph. We evaluate the interest of the study of the numerical range of non self-adjoint operators. Indeed, the knowledge of the numerical range of the Laplacian brings an essential information on its essential spectrum.
We provide the Cheeger inequality at infinity on a filtration of graphG.
Definition 3.12. A graph H =(VH, ~EH)is called a subgraph of G=(VG, ~EG)if VH⊂VGandE~H=n
(x,y); x,y∈ VH
o∩E~G.
Definition 3.13. A filtration of G=(V, ~E)is a sequence of finite connected subgraphs{Gn=(Vn, ~En), n∈N}such that Gn⊂Gn+1and:
[
n≥1
Vn=V.
LetGbe an infinite connected graph and{Gn, n∈N}a filtration ofG. Let us denote m∞= lim
n→∞ mVnc
M∞= lim
n→∞ MVcn
The Cheeger constant at infinity is defined by:
h∞= lim
n→∞h(Vcn).
Remark 3.14. These limits exist inR+∪ {∞}because mVnc, MVcnand h(Vcn)are monotone sequences.
Remark 3.15. The Cheeger constant at infinity h∞is independent of the filtration. Indeed it can be defined, as in [9]
and [15], by h∞= lim
K→Gh(Kc), where K runs over all finite subsets because the graph is locally finite.
Definition 3.16. G is called with heavy ends if m∞ =∞.
Lemma 3.17. For any subsetΩof V such thatΩcis finite, we have ν(∆DΩ)=ν(∆DΩ).
Proof. It is easy to see that
b= inf
λ∈W(∆DΩ)
Re(λ)≤ inf
λ∈W(∆DΩ)
Re(λ)=a.
Let f ∈ D(∆DΩ) = {f ∈ D(∆), f(x) = 0, ∀ x ∈ Ωc} such that k f km= 1. Hence there is a sequence (fn)∈ Cc(V)=D(∆) which converges to f and (∆fn) converges to∆f. It follows that1n =1Ωfn =0 onΩc and (∆DΩ1n) converges to∆DΩf. So
a≤ν(∆DU)≤ Re(∆DU1n,1n)m −→
n→∞
Re(∆DUf,f)m
then
a≤b.
Theorem 3.18. The essential spectrum of∆satisfies:
h2∞
8 ≤ M∞ηess(∆) and
m∞
h˜2∞
8 ≤ ηess(∆). (8)
Proof. Let{Gn, n∈N}be a filtration ofG, from the inequality (3) we get, ν(∆DVcn)≤ηess(∆DVcn).
From Theorem 5.35 of T. Kato page 244 [14], the essential spectrum is stable by a compact perturbation, we obtain
σess(∆)=σess(∆DVnc).
Therefore
ν(∆DVcn)≤ηess(∆),
we use Theorem 3.7 and the equality (6), then we find the result by taking the limit at∞.
The following Corollary follows from Theorem 3.18. It gives an important characterization for the absence of the essential spectrum especially it includes the case of rapidly branching graphs.
Corollary 3.19. The essential spectrum of∆on a heavy end graph G withh˜∞>0is empty.
Proof. The emptiness of the essential spectrum for∆on a graph withheavy endsis an immediate Corollary of the inequality (8), then ifm∞=∞where ˜h∞ >0, we haveσess(∆)=∅.
Acknowledgments: I take this opportunity to express my gratitude to my thesis directors Colette Ann´e and Nabila Torki-Hamza for all the fruitful discussions, helpful suggestions and their guidance during this work. I would like to thank the anonymous referee for the careful reading of my paper and the valuable comments and suggestions.
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